but it can't be done

But It Can't Be DoneAuthor(s): Larry P. Leutzinger and Glenn NelsonSource: The Arithmetic Teacher, Vol. 27, No. 8 (April 1980), pp. 6-9Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191724 .

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Leti Do léL" By Larry R Leutzinger Area Education Agency 7 Cedar Falls, Iowa and Glenn Nelson University of Northern Iowa Cedar Falls, Iowa

Teachers in elementary classrooms throughout the country often hear students say, "I can't do it." The state- ment does not imply that the problem or exercise being attempted has no solution but, rather, that the student does not have the skills necessary to com- plete it, or that the student has forgot- ten how to determine the answer.

The intent of this month's column, however, is to present problems and activities for which "I can't do it" or "It can't be done" are the correct re- sponses. Including problems of this type in the curriculum adds a new di- mension to students' learning and helps to develop better problem solvers. The ability to see that a problem has no solution often requires much more insight than simply deter- mining the answer to a solvable problem.

The "unsolvable" problems will fall

into four categories: story problems with missing information; compu- tational problems that appear unsolvable to the students, but with the development of additional skills will be solvable; activities which, though unsolvable, develop concepts; and open-ended activities.

Story Problems with Information Missing Problems of this type may require the students to decide what information is missing, to supply reasonable data, and then to answer the question. The following problems are examples.

1. Martin selects one box of pencils for $1.45, two erasers for 48Ф, and a pad of paper for 96Ф. How much change will he receive from the check- out clerk when he pays for these items?

2. How many cars will be needed to

transport 39 children on a field trip?

3. Juan and Maria want to make scarves for their cousins. They have 3.2 meters of material. How long will each scarf be if the length of each scarf is the same?

A problem may also include unneces- sary information.

4. Debra lives at 1979 Hawthorne Street. Her sister Cindy is eight years old. How old will Debra be in ten years?

If problems of this type are included in each assignment involving story problems, the students will be required to think carefully. They will be less likely to plug numbers randomly into formulas or algorithms. Teachers can use problems in books, with some information removed, and ask the students to supply their own data.

6 Arithmetic Teacher



Computations That Can't Be Done - Yet

These "unsolvable" computations may be used in more than one way. They can be presented prior to the introduction of a new topic to provide a rationale for that topic. They can also serve to emphasize a particular aspect of a topic. Some examples follow.

1. Circle the problems that cannot be solved.

8 6 9 3 5 6 2 -4 -2 -7 -5 -7 -1 -8

For first or second graders, this activity focuses attention on the concept of subtraction and the reading of subtraction examples. This can be a valu- able activity immediately prior to studying regrouping in subtraction.

For sixth graders, this activity can

serve as an introduction to integers. 2. Circle the problems that cannot be

solved.

6Î48 8)57 3)24 9)85 12Î7

Examples like these can be used to review the multiplication facts or to develop a need for expressing remainders in fractional or decimal form.

3. Circle the expressions that are not proper fractions.

3 7 0 5 4 4 6 3 0 1

Definitions can be introduced, re- fined, and broadened through the use of counter examples like these. In these five representations you have examples not only of proper fractions but also of improper fractions and one non- fraction. The definition of a fraction can move from parts of a whole, to a

ratio of whole numbers, to an indicated quotient, to rational numbers, depend- ing on the level of your students.

Problems with No Solutions Some problems that have no solution are useful in developing concepts. De- termining if and why a problem has no solution requires a student to think very carefully. In each of the following activities, two of the three problems have solutions, in some cases more than one solution. In the third problem, no solution is possible and the students are to indicate how they know this.

Greater than and less than 1. What whole number is between 15

and 17? 2. What number is greater than 20 and

less than 42?

April 1980 7

ä Dut к Can't ama n PN w n По PN Done



3. What number is greater than 42 and less than 20?

The first problem has one solution, 16. The second has twenty-one possible solutions. The third has no solution be- cause a number cannot be greater than 42 and less than 20.

Odd and even numbers 1. Find two consecutive numbers

whose sum is 15. 2. Find three odd numbers whose sum

is 17. 3. Find four odd numbers whose sum

is 23.

The first problem has one solution, 7 and 8. The second has several solutions - 1 +7 + 9, 3 + 5 + 9, 3 + 7 + 7, and so on. The third example has no solution since the sum of four odd numbers will always be an even number. Additional problems of this type can be created to investigate concepts of products and differences of even and odd numbers.

Prime numbers

1. Express 26 as the sum of two prime numbers.

2. Express 17 as the sum of three prime numbers.

3. Express 23 as the sum of two prime numbers.

The first problem has several solutions (13 + 13, 23 + 3, 19 + 7), which can be determined by trial and error. The second has several solutions (13+1+3, 7 + 5 + 5, 1 1 + 3 + 3, and so on). The third problem has no solution. To have a solution, one of the primes must be even and the other odd in order that the sum be odd. The only even prime number is 2. But 2 + 21 is equal to 23, and 21 is not a prime number.

Square numbers 1. Find two consecutive numbers

whose squares differ by 1 1. 2. Find two consecutive numbers

whose squares differ by 79. 3. Find two consecutive numbers

whose squares differ by 100.

The solution to the first problem can be easily determined (5 and 6). The second problem is not quite so easily solved, but the answer (39 and 40) can

be determined by trial and error. In the third example, a trial-and-error ap- proach will soon lead to frustration. If consecutive numbers are listed and their squares determined, it is soon ap- parent that the squares of consecutive numbers always differ by an odd number.

Open-ended Activities The following activities present unsolvable situations from which the students can develop generalizations concerning the concepts involved.

First activity The students are given four sets of paper strips:

Set A Three, 6-cm-by- 1 -cm strips Set В Two, 6-cm-by- 1 -cm strips;

one 3-cm-by-l-cm strip Set С One, 6-cm-by- 1 -cm strip;

two, 3-cm-by-l-cm strips Set D One, 6-cm-by- 1 -cm strip;

one, 3-cm-by-l-cm strip; one, 2-cm-by-l-cm strip

They are to take each set of strips and attempt to form a triangle with the strips in that set. With which of the sets of strips can triangles be formed? (A and B) Why can't triangles be formed using the other sets of strips? What must be true of the three strips before a triangle can be formed? (The sum of the lengths of any two strips must be longer than the length of the other strip.)

A list of the lengths of strips in five other sets of three strips can then be given.

1. 7 cm, 8 cm, 9 cm 2. 12 cm, 12 cm, 12 cm 3. 1 cm, 1 cm, 6 cm 4. 15 cm, 15 cm, 1 cm 5. 5 cm, 5 cm, 11 cm

With which of the listed sets of strips can triangles be formed?

To extend the activity, give each student an additional 14-cm-by-l-cm strip. The students are to attempt to form a quadrilateral using the three strips in each set (A, B, C, and D) and the M-cm-by-l-cm strip as the sides of the quadrilateral. With which sets of strips can a quadrilateral be formed?

(A and B) What must be true of the four strips before a quadrilateral can be formed? (The combined lengths of any three strips must be longer than the other strip.) What size strip could be added to the three strips in set D so that a quadrilateral could be formed. (3 cm) What size strip could be added to set A with the result that a quadrilateral could not be formed?

Exercises listing the lengths of four strips and requiring the students to determine which set of four strips could form a quadrilateral might be devel- oped. The activity could be extended further by adding another strip and asking the students to form pentagons. Generalizations regarding the lengths of sides of pentagons could then be de- veloped.

Second activity The students will need graph paper. The sides of the figures that the students draw in the first two activities must lie on the grid lines of the graph paper. The length of a square of the graph paper will be one unit.

Perimeters on graph paper Have students draw rectangles with the following perimeters: 12 units, 18 units, 15 units, 8 units, and 21 units. What conclusions can be drawn about the perimeters of the rectangles that have been drawn? (Each perimeter is an even number of units.)

Have students draw squares with the following perimeters: 12 units, 10 units, 16 units, 26 units, 3 units, and 20 units. What conclusions can be made about the perimeters of squares drawn on the graph paper? (The perimeters must be a multiple of four units.)

Areas on graph paper Have students draw rectangles that have the following areas (one square on the graph paper has an area of one square unit): 1 square unit, 5 square units, 9 square units, 6 square units, and 12 square units. Is it possible to make rectangles of any area provided the graph paper is large enough?

Have students draw squares that have the following areas: 4 square units, 10 square units, 16 square units, 15 square units, 25 square units. Is it possible to make squares of any area

8 Arithmetic Teacher



Fig. 1

Fig. 2

provided the graph paper is large enough? For what areas can squares be made? (1, 4, 9, 16, and so on)

Challenging areas The drawing of squares and rectangles on graph paper can be extended to figures whose sides do npt lie on the grid lines of the graph paper, but the vertices of the figures are on an intersection of two grid lines. For these activities students need to know how to find the areas of triangular regions or to use the Pythagorean theorem, and how to determine the areas of square and rectangular regions that are not vertically and horizontally aligned. (See fig. 1)

Have students draw squares with the following areas (the sides of the squares need not lie on grid lines, but the vertices of the squares must lie on the intersection of two grid lines): 1 square unit, 2 square units, 3 square units, 4 square units, 5 square units, 6 square units, 7 square units, 8 square units, 9 square units, and 10 square units. For which of these areas is it possible to draw squares? (1, 2, 4, 5, 8, 9, and 10 square units) Why is it possible to make squares with these areas? The answer, simply put, is that these areas can be expressed as the sum of two squares. (See fig. 2) For what other areas can squares be drawn on graph paper? (13, 16, 17, 20, 23, and so on)

As a further extension, have students examine the areas of rectangles drawn with the vertices of the rectangles on the intersections of grid lines of graph paper.

Summary The purpose of this month's column has been to show how problems that have no solution or more than one solution can be used in teaching mathematics to develop thinking skills, expand topics, and encourage concept development. The sample problems of these types that have been included here should give teachers examples to use and expand upon in their own classrooms. The continued use of problems of these types could well result in the development of discriminating problem solvers.D

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but it can't be done

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